Soggetto

Abstract

Given a nonparametric regression model, we assume that the number of
covariates d → ∞ but only some of these covariates are relevant for the model. Our goal
is to identify the relevant covariates and to obtain some information about the structure of
the model. We propose a new nonparametric procedure, called GRID, having the following
features: (a) it automatically identifies the relevant covariates of the regression model, also
distinguishing the nonlinear from the linear ones (a covariate is defined linear/nonlinear
depending on the marginal relation between the response variable and such a covariate);
(b) the interactions between the covariates (mixed effect terms) are automatically identified,
without the necessity of considering some kind of stepwise selection method. In
particular, our procedure can identify the mixed terms of any order (two way, three way,
...) without increasing the computational complexity of the algorithm; (c) it is completely
data-driven, so being easily implementable for the analysis of real datasets. In particular,
it does not depend on the selection of crucial regularization parameters, nor it requires the
estimation of the nuisance parameter 2 (self scaling). The acronym GRID has a twofold
meaning: first, it derives from Gradient Relevant Identification Derivatives, meaning that
the procedure is based on testing the significance of a partial derivative estimator; second,
it refers to a graphical tool which can help in representing the identified structure of the
regression model. The properties of the GRID procedure are investigated theoretically.

Given a nonparametric regression model, we assume that the number of
covariates d → ∞ but only some of these covariates are relevant for the model. Our goal
is to identify the relevant covariates and to obtain some information about the structure of
the model. We propose a new nonparametric procedure, called GRID, having the following
features: (a) it automatically identifies the relevant covariates of the regression model, also
distinguishing the nonlinear from the linear ones (a covariate is defined linear/nonlinear
depending on the marginal relation between the response variable and such a covariate);
(b) the interactions between the covariates (mixed effect terms) are automatically identified,
without the necessity of considering some kind of stepwise selection method. In
particular, our procedure can identify the mixed terms of any order (two way, three way,
...) without increasing the computational complexity of the algorithm; (c) it is completely
data-driven, so being easily implementable for the analysis of real datasets. In particular,
it does not depend on the selection of crucial regularization parameters, nor it requires the
estimation of the nuisance parameter 2 (self scaling). The acronym GRID has a twofold
meaning: first, it derives from Gradient Relevant Identification Derivatives, meaning that
the procedure is based on testing the significance of a partial derivative estimator; second,
it refers to a graphical tool which can help in representing the identified structure of the
regression model. The properties of the GRID procedure are investigated theoretically.